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Christoph Walker - One of the best experts on this subject based on the ideXlab platform.

  • a free Boundary Problem modeling electrostatic mems ii nonlinear bending effects
    Mathematical Models and Methods in Applied Sciences, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    Well-posedness of a free Boundary Problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic plate suspended above a fixed ground plate together with the electrostatic potential in the free domain between the two plates. The electrostatic potential is harmonic in that domain and its values are held fixed along each plate. The equation for the elastic plate deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the elastic plate.

  • a variational approach to a stationary free Boundary Problem modeling mems
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    A variational approach is employed to find stationary solutions to a free Boundary Problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization Problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.

  • a free Boundary Problem modeling electrostatic mems i linear bending effects
    Mathematische Annalen, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    The dynamical and stationary behaviors of a fourth-order evolution equation with clamped Boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free Boundary Problem in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems and includes two positive parameters $$\lambda $$ and $$\varepsilon $$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution Problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small  $$\lambda $$ . Non-existence of steady states is also established when $$\varepsilon $$ is small enough and $$\lambda $$ is large enough (depending on $$\varepsilon $$ ).

  • A Parabolic Free Boundary Problem Modeling Electrostatic MEMS
    Archive for Rational Mechanics and Analysis, 2014
    Co-Authors: Joachim Escher, Philippe Laurençot, Christoph Walker
    Abstract:

    The evolution Problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the Boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free Boundary Problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

  • A free Boundary Problem modeling electrostatic MEMS: II. nonlinear bending effects
    Mathematical Models and Methods in Applied Sciences, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    Well-posedness of a free Boundary Problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fixed ground plate. The electrostatic potential is harmonic in that domain and its values are held fixed along the membrane and the ground plate. The equation for the membrane deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the membrane.

Tao Luo - One of the best experts on this subject based on the ideXlab platform.

  • ill posedness of free Boundary Problem of the incompressible ideal mhd
    Communications in Mathematical Physics, 2020
    Co-Authors: Chengchun Hao, Tao Luo
    Abstract:

    In the present paper, we show the ill-posedness of the free Boundary Problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $$\mu _0$$, in Sobolev spaces. The analysis is uniform for any $$\mu _0>0$$.

  • a priori estimates for free Boundary Problem of incompressible inviscid magnetohydrodynamic flows
    Archive for Rational Mechanics and Analysis, 2014
    Co-Authors: Chengchun Hao, Tao Luo
    Abstract:

    In the present paper, we prove the a priori estimates of Sobolev norms for a free Boundary Problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536‐1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition thattheouternormalderivativeofthetotalpressureincludingthefluidandmagnetic pressuresisnegativeonthefreeBoundary,whichissimilartothephysicalcondition (Taylor sign condition) for the incompressible Euler equations of fluids.

  • a priori estimates for free Boundary Problem of incompressible inviscid magnetohydrodynamic flows
    arXiv: Analysis of PDEs, 2013
    Co-Authors: Chengchun Hao, Tao Luo
    Abstract:

    In the present paper, we prove the a priori estimates of Sobolev norms for a free Boundary Problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free Boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.

Philippe Laurençot - One of the best experts on this subject based on the ideXlab platform.

  • a free Boundary Problem modeling electrostatic mems ii nonlinear bending effects
    Mathematical Models and Methods in Applied Sciences, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    Well-posedness of a free Boundary Problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic plate suspended above a fixed ground plate together with the electrostatic potential in the free domain between the two plates. The electrostatic potential is harmonic in that domain and its values are held fixed along each plate. The equation for the elastic plate deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the elastic plate.

  • a variational approach to a stationary free Boundary Problem modeling mems
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    A variational approach is employed to find stationary solutions to a free Boundary Problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization Problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.

  • a free Boundary Problem modeling electrostatic mems i linear bending effects
    Mathematische Annalen, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    The dynamical and stationary behaviors of a fourth-order evolution equation with clamped Boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free Boundary Problem in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems and includes two positive parameters $$\lambda $$ and $$\varepsilon $$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution Problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small  $$\lambda $$ . Non-existence of steady states is also established when $$\varepsilon $$ is small enough and $$\lambda $$ is large enough (depending on $$\varepsilon $$ ).

  • A Parabolic Free Boundary Problem Modeling Electrostatic MEMS
    Archive for Rational Mechanics and Analysis, 2014
    Co-Authors: Joachim Escher, Philippe Laurençot, Christoph Walker
    Abstract:

    The evolution Problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the Boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free Boundary Problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

  • A free Boundary Problem modeling electrostatic MEMS: II. nonlinear bending effects
    Mathematical Models and Methods in Applied Sciences, 2014
    Co-Authors: Philippe Laurençot, Christoph Walker
    Abstract:

    Well-posedness of a free Boundary Problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fixed ground plate. The electrostatic potential is harmonic in that domain and its values are held fixed along the membrane and the ground plate. The equation for the membrane deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the membrane.

Chengchun Hao - One of the best experts on this subject based on the ideXlab platform.

Adriana C Briozzo - One of the best experts on this subject based on the ideXlab platform.

  • on a non linear moving Boundary Problem for a diffusion convection equation
    International Journal of Non-linear Mechanics, 2012
    Co-Authors: Adriana C Briozzo, Maria Fernanda Natale
    Abstract:

    Abstract We study a one-dimensional free Boundary Problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Backlund transformation the Problem is reduced to an associated free Boundary Problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.